On the small oscillations of the periodic Rayleigh equation
Author:
Hal L. Smith
Journal:
Quart. Appl. Math. 44 (1986), 223-247
MSC:
Primary 34C15; Secondary 58F14, 58F15, 92A15
DOI:
https://doi.org/10.1090/qam/856177
MathSciNet review:
856177
Full-text PDF Free Access
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V. I. Arnold, Loss of stability of self-oscillations close to resonances and versal deformations of equivariant vector fields, Funct. Anal. Appl. 11, 1–10 (1977)
V. I. Arnold, Geometrical methods in the theory of ordinary differential equations, Springer-Verlag, 1983
F. S. Berezovskaia and A. I. Khibnik, On the bifurcation of separatrices in the problem of stability loss of auto-oscillations near 1:4 resonance, P.M M. 44, 663–667 (1981)
J. Carr, Applications of center manifold theory, Springer-Verlag, 1981
S. N. Chow, J. K. Hale and J. Mallet-Paret, An example of bifurcation to homoclinic orbits, J. Differential Equations 37, 351–373 (1980)
J. Guckenheimer and P. Holmes, Nonlinear oscillations, dynamical systems, and bifurcations of vector fields, Springer-Verlag, New York, 1983
J. K. Hale, Ordinary differential equations, Wiley-Interscience, New York, 1969
M. R. Herman, Mesure de Lebesgue et nombre de rotation, in Geometry and Topology, J. Palis and M. de Carmo (eds.), Lect. Notes in Math., Vol. 597, Springer-Verlag, 271–293 (1977)
P. J. Holmes, Averaging and chaotic motions in forced oscillations, Siam J. Appl. Math. 38, 65–80 (1980)
E. I. Horozov, Versal deformations of equivariant vector fields for the cases of symmetries of order 2 and 3, Trudy Sem. I. G. Petrovskoyo 5, 163–192 (1979)
G. Iooss, Bifurcation of maps and applications, Math. Studies, Vol. 36, North Holland, Amsterdam
W. Magnus and S. Winkler, Hill’s equation, Interscience, New York, 1966
V. K. Melnikov, On the stability of the center for time periodic perturbations, Trans. Moscow Math. Soc. 12, 1–57 (1963)
A. I. Neishtadt, Bifurcations of the phase pattern of an equation system arising in the problem of stability loss of self-oscillations close to 1:4 resonance, P.M.M. 42, 830–840 (1978)
B. Simon, On the genericity of nonvanishing instability intervals, Ann. Inst. H. Poincaré, A 24, 91 (1976)
H. L. Smith, Dynamic bifurcation in a perturbed Hill’s equation, preprint
H. L. Smith, Normal forms for periodic systems, preprint
J. J. Stoker, Nonlinear vibrations, Wiley, New York, 1950
F. Takens, Normal forms for certain singularities of vector fields, Ann. Inst. Fourier 23, 163–195 (1973)
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Article copyright:
© Copyright 1986
American Mathematical Society